Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$. (a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular. (Walther Janous)